The filtered differential induces d r on the r page

For r=0 use [F2]. For r≥1, the restriction of d to $A_{p,n}^r$ lands in $F_{p-r}{C}_{n-1}$. Its next differential is zero, so it lands in $A_{p-r,n-1}^r$, the target numerator in [F1].

If $x$ changes by $a\in A_{p-1,n}^{r-1}$, then $da$ lies in the target boundary summand $d(A_{(p-r)+r-1,n}^{r-1})=d(A_{p-1,n}^{r-1})$. If x changes by $db$ with $b\in A_{p+r-1,n+1}^{r-1}$, its differential changes by $d^{2}b=0$. These are subobject factorization calculations: the sum epimorphism allows their verification after an epic cover by [F3].

Consequently the map from the source numerator to the target quotient kills both denominator summands and hence their sum. Cokernel descent [F3] gives a unique $d^r$ satisfying the stated rule. Equality after the epic source quotient proves that the rule is independent of every permitted local representative.